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We study a driven-dissipative model of spins one-half (qubits) on a latticewith nearest-neighbor interactions. Focusing on the role of spatially extendedspin-spin correlations in determining the phases of the system, we characterizethe spatial structure of the correlations in the steady state, as well as theirtemporal dynamics. In dimension one we use essentially exactmatrix-product-operator simulations on large systems, and pushing thesecalculations to dimension two, we obtain accurate results on small cylinders.We also employ an approximation scheme based on solving the dynamics of themean field dressed by the feedback of quantum fluctuations at leading order.This approach allows us to study the effect of correlations in large latticeswith over one hundred thousand spins, as the spatial dimension is increased upto five. In dimension two and higher we find two new states that are stabilizedby quantum correlations and do not exist in the mean-field limit of the model.One of these is a steady state with mean magnetization values that lie betweenthe two bistable mean-field values, and whose correlation functions haveproperties reminiscent of both. The correlation length of the new phasediverges at a critical point, beyond which we find emerging a new limit cyclestate with the magnetization and correlators oscillating periodically in time.

Correlation-induced steady states and limit cycles in driven dissipative quantum systems